Answer:

### Find the vector equation of the plane through the points and parallel to the plane .

Answer:

### Show that the planes 2x – y + 6z = 5 and 5x – 2.5y + 15z = 12 are parallel.

Answer:

### Find the vector and Cartesian equations of a plane which is at a distance of 7 units from the origin and whose normal vector from the origin is

Answer: Given, $\begin{array}{l} d = 7\\ \overline n = 3\widehat i + 5\widehat j - 6\widehat k \end{array}$ The unit vector normal to the plane: ...

### Find the distance of the point (2, 1, 0) from the plane 2x + y – 2z + 5 = 0.

Answer: Given plane, 2x + y – 2z + 5 = 0 The point is (2, 1, 0).

### Find the distance of the point (1, 1, 2) from the plane plane .

Answer: Given plane, $\overrightarrow r .(2\widehat i - 2\widehat j + 4\widehat k) + 5 = 0$ The cartesian form: 2x – 2y + 4z + 5 = 0 The point is (1, 1, 2).

### Find the distance of the point (3, 4, 5) from the plane .

Answer: Given plane, $\overrightarrow r .(2\widehat i - 5\widehat j + 3\widehat k) = 13$ The cartesian form: 2x – 5y + 3z – 13 = 0 The point is (3, 4, 5).

### Find the distance of the point from the plane .

Answer: Given plane, $\overrightarrow r .(\widehat i + \widehat j + \widehat k) + 17 = 0$ The cartesian form: x + y + z +17 = 0 The point is $(\widehat i + 2\widehat j + 5\widehat k)$ => (1, 2,...

### Find the distance of the point from the plane .

Answer: Given plane, $\overrightarrow r .(3\widehat i - 4\widehat j + 12\widehat k) = 9$ The cartesian form: 3x – 4y + 12z – 9 =0 The point is $(2\widehat i - \widehat j - 4\widehat k)$ => (2,...

### Find the vector and Cartesian equations of a plane which passes through the point (1, 4, 6) and normal vector to the plane is is

Answer: Given, A = (1, 4, 6)

### Find the vector and Cartesian equations of a plane which is at a distance of 6 units from the origin and which has a normal with direction ratios 2, -1, -2.

Answer: Given, d = 6

### Find the vector and Cartesian equations of a plane which is at a distance of 6/√29 from the origin and whose normal vector from the origin is is

Answer: Given, = (x × 2) + (y × (-3)) + (z × 4) = 2x - 3y + 4z The Cartesian equation...

### Find the vector equation of a plane which is at a distance of 5 units from the origin and which has as the unit vector normal to it.

Answer: Given, $\begin{array}{l} d = 5\\ \widehat n = \widehat k \end{array}$ The equation of plane at 5 units distance from the origin $\begin{array}{l} \widehat n \end{array}$ and as a unit...

### Reduce the equation of the plane 4x – 3y + 2z = 12 to the intercept form, and hence find the intercepts made by the plane with the coordinate axes.

Answer: Equation of the plane: 4x – 3y + 2z = 12 $\begin{array}{l} \frac{4}{{12}}x - \frac{3}{{12}}y + \frac{2}{{12}}z = 1\\ \frac{x}{3} + \frac{y}{{ - 4}} + \frac{z}{6} = 1 \end{array}$ It is the...

### Write the equation of the plane whose intercepts on the coordinate axes are 2, – 4 and 5 respectively.

Answer: Given, Coordinate axes are 2, - 4, 5 The equation of the variable plane: The required equation of the plane is 10x – 5y + 4z = 20.

### Find the equation of the plane passing through each group of points: A (-2, 6, -6), B (-3, 10, -9) and C (-5, 0, -6)

Answer: Given, A (-2, 6, -6) B (-3, 10, -9) C (-5, 0, -6) ...

### 3. Show that the four points A (0, -1, 0), B (2, 1, -1), C (1, 1, 1) and D (3, 3, 0) are coplanar. Find the equation of the plane containing them.

Answer: Given, A (0, -1, 0) B (2, 1, -1) C (1, 1, 1) D (3, 3, 0) 4x – 3 (y + 1) + 2z = 0 4x – 3y + 2z – 3 = 0 Take x = 0, y = 3 and z =...

### Show that the four points A (3, 2, -5), B (-1, 4, -3), C (-3, 8, -5) and D (-3, 2, 1) are coplanar. Find the equation of the plane containing them.

Answer: Let us take, The equation of the plane passing through A (3, 2, -5) a (x – 3) + b (y – 2) + c (z + 5) = 0 It passes through the points B (-1, \4, -3) and C (-3, 8, -5) a (1 – 3) + b (4 – 2)...

### Find the equation of the plane passing through each group of points: (i) A (2, 2, -1), B (3, 4, 2) and C (7, 0, 6) (ii) A (0, -1, -1), B (4, 5, 1) and C (3, 9, 4)

Answer: (i) Given, A (2, 2, -1) B (3, 4, 2) C (7, 0, 6) ...

### Find the length of tangent drawn to a circle with radius 8 cm form a point 17 cm away from the center of the circle

### Using properties of determinants prove that:

Solution: $=\left|\begin{array}{lll} b(b-a) & b-c & c(b-a) \\ a(b-a) & a-b & b(b-a) \\ c(b-a) & c-a & a(b-a) \end{array}\right|$ Taking (b-a) common from $\mathrm{C}_{1},...

### Using properties of determinants prove that:

Solution: Expanding with R1 $\begin{array}{l} =b^{2} c^{2}\left(a^{2} c+a b c-a b c-a^{2} b\right)-b c\left(a^{3} c^{2}+a^{2} b c^{2}-a^{2} b^{2} c-a^{3} b^{2}\right)+(b+c)\left(a^{3} b c^{2}-a^{3}...

### Using properties of determinants prove that:

Solution: $\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \\ (a-1)^{2} & (b-1)^{2} & (c-1)^{2}\end{array}\right|$...

### Using properties of determinants prove that:

Solution: $\left|\begin{array}{ccc} a & b & a x+b y \\ b & c & b x+c y \\ a x+b y & b x+c y & 0 \end{array}\right|$ $\begin{array}{l} \left.=\left(\frac{1}{x...

### Using properties of determinants prove that:

Solution: $\begin{array}{l} \left|\begin{array}{ccc} \mathrm{b}+\mathrm{c} & \mathrm{a} & \mathrm{a} \\ \mathrm{b} & \mathrm{c}+\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{c}...

### Using properties of determinants prove that:

Solution: $\begin{array}{l} \left|\begin{array}{ccc} a^{2}+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1 \end{array}\right| \\ =\left|\begin{array}{ccc} a^{2}-1 & a-1...

### Using properties of determinants prove that:

Solution: $\left|\begin{array}{ccc}x+4 & 2 x & 2 x \\ 2 x & x+4 & 2 x \\ 2 x & 2 x & x+4\end{array}\right|$ $=\left|\begin{array}{ccc}5 \mathrm{x}+4 & 5 \mathrm{x}+4...

### Evaluate :

Solution: $\begin{array}{l} \left|\begin{array}{lll} 67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26 \end{array}\right| \\...

### If is a matrix such that and then write the value of .

Solution: Theorem: If Let $A$ be $k \times k$ matrix then $|p A|=p^{k}|A|$. Given: $\mathrm{k}=3$ and $\mathrm{p}=3$. $\begin{array}{l} |3 \mathrm{~A}|=3^{3} \times|\mathrm{A}| \\ =27|\mathrm{~A}|...

### Prove that

Answer:

### Prove that

Answer: = cosx cos2x cos4x cos8x Multiply and divide by 2sinx, we get We know that, sin 2x = 2 sinx cosx Replacing x by 2x, we get sin 2(2x) = 2 sin(2x) cos(2x) or sin 4x = 2 sin 2x cos 2x...

### Prove that

Answer: Taking sinx common from the numerator and cosx from the denominator

### Prove that:

Answer: = RHS ∴ LHS = RHS Hence Proved

### Prove that:

Answer: Multiply and divide by 2, we get = RHS ∴ LHS = RHS Hence Proved

### Prove that cot x – 2cot 2x = tan x

Answer: Taking LHS, = cot x – 2cot 2x …(i) We know that, cot x = cos x/ sin x Replacing x by 2x, we get cot 2x = cos 2x/ sin 2x So, eq. (i) becomes = sin x/ cos x = tan x = RHS ∴ LHS = RHS Hence...

### Prove that cos 2x + 2sin2 x = 1

Answer: Taking LHS = 2 – 1 = 1 = RHS

### Prove that cosec 2x + cot 2x = cot x

Answer: To Prove: cosec 2x + cot 2x = cot x Taking LHS, = cosec 2x + cot 2x …(i) We know that, cosec x = 1 / sin x and cot x = cos x/ sin x Replacing x by 2x, we get = cos x/ sinx = cot...

### Prove that sin 2x(tan x + cot x) = 2

Answer: Taking LHS, sin 2x(tan x + cot x) We know that: We know that, sin 2x = 2 sinx cosx = 2 = RHS ∴ LHS = RHS Hence Proved

### Prove that:

Answer: = sin x / cos x = tan x = RHS ∴ LHS = RHS Hence Proved

### In a four-sided field, the length of the longer diagonal is 128 m. The lengths of perpendiculars from the opposite vertices upon this diagonal are 22.7 m and 17.3 m. Find the area of the field.

### The adjacent sides of a parallelogram are 36 cm and 27 cm in length. If the distance between the shorter sides is 12 cm, find the distance between the longer sides.

### The diagonals of a rhombus are 48 cm and 20 cm long. Find the perimeter of the rhombus.

### A parallelogram and a rhombus are equal in area. The diagonals of the rhombus measure 120 m and 44 m. If one of the sides of the || gm is 66 m long, find its corresponding altitude.

### The cost of fencing a square lawn at 14 per metre is 2800. Find the cost of mowing the lawn at ₹ 54 per 100 m2.

### The adjacent sides of a ||gm ABCD measure 34 cm and 20 cm and the diagonal AC is 42 cm long. Find the area of the ||gm.

### Find the area of a trapezium whose parallel sides are 11 cm and 25 cm long and non- parallel sides are 15 cm and 13 cm.

### Find the area of a rhombus each side of which measures 20 cm and one of whose diagonals is 24 cm.

### A lawn is in the form of a rectangle whose sides are in the ratio 5:3 and its area is Find the cost of fencing the lawn at ₹ 20 per metre.

### Find the area of a triangle whose sides are 42 cm, 34 cm and 20 cm.

### Find the area of a rhombus whose diagonals are 48 cm and 20cm long.

### The length of the diagonal of a square is 24 cm. Find its area.

### The longer side of a rectangular hall is 24 m and the length of its diagonal is 26 m. Find the area of the hall.

### Find the area of an isosceles triangle each of whose equal sides is 13 cm and whose base is 24 cm.

### Find the area of an equilateral triangle having each side of length 10 cm. (Take

### The parallel sides of a trapezium are 9.7cm and 6.3 cm, and the distance between them is 6.5 cm. The area of the trapezium is (a) 104 cm2 (b) 78 cm2 (c) 52 cm2 (d) 65 cm2

### The sides of a triangle are in the ratio 12: 14 : 25 and its perimeter is 25.5 cm. The largest side of the triangle is (a) 7 cm (b) 14 cm (c) 12.5 cm (d) 18 cm

### In the given figure ABCD is a trapezium in which AB =40 m, BC=15m,CD = 28m, AD= 9 m and CE = AB. Area of trapezium ABCD is

### In the given figure ABCD is a quadrilateral in which

### Find the area of trapezium whose parallel sides are 11 m and 25 m long, and the nonparallel sides are 15 m and 13 m long.

### The shape of the cross section of a canal is a trapezium. If the canal is 10 m wide at the top, 6 m wide at the bottom and the area of its cross section is 640 m2 , find the depth of the canal.

### The parallel sides of trapezium are 12 cm and 9cm and the distance between them is 8 cm. Find the area of the trapezium.

### The area of rhombus is 480 c m2 , and one of its diagonal measures 48 cm. Find

(i) the length of the other diagonal,

(ii) the length of each of the sides

(iii) its perimeter

### The perimeter of a rhombus is 60 cm. If one of its diagonal us 18 cm long, find (i) the length of the other diagonal, and (ii) the area of the rhombus.

### Find the area of the rhombus, the length of whose diagonals are 30 cm and 16 cm. Also, find the perimeter of the rhombus.

### The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm, and the diagonal AC measures 42 cm. Find the area of the parallelogram.

### The area of a parallelogram is 392 m2 . If its altitude is twice the corresponding base, determined the base and the altitude.

### The adjacent sides of a parallelogram are 32 cm and 24 cm. If the distance between the longer sides is 17.4 cm, find the distance between the shorter sides.

### Find the area of a parallelogram with base equal to 25 cm and the corresponding height measuring 16.8 cm.

Sol: Given: Base = 25 cm Height = 16.8 cm \Area of the parallelogram = Base ´ Height = 25cm ´16.8 cm = 420 cm2

### Find the area of the quadrilateral ABCD in which in AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm and diagonal BD = 20 cm.

### Find the perimeter and area of the quadrilateral ABCD in which AB = 17 cm, AD = 9 cm, CD = 12 cm, ACB 90 and AC = 15 cm.

### Find the area of the quadrilateral ABCD in which AD = 24 cm, BAD 90 and BCD is an equilateral triangle having each side equal to 26 cm. Also, find the perimeter of the quadrilateral. Sol:

### In the given figure ABCD is quadrilateral in which diagonal BD = 24 cm, AL BD and CM BD such that AL = 9cm and CM = 12 cm. Calculate the area of the quadrilateral.

### The cost of fencing a square lawn at ₹ 14 per meter is ₹ 28000. Find the cost of mowing the lawn at ₹ 54per100 m2

### The cost of harvesting a square field at ₹ 900 per hectare is ₹ 8100. Find the cost of putting a fence around it at ₹ 18 per meter.

### The area of a square filed is 8 hectares. How long would a man take to cross it diagonally by walking at the rate of 4 km per hour?

### Find the length of the diagonal of a square whose area is 128 cm2 . Also, find its perimeter.

### Find the area and perimeter of a square plot of land whose diagonal is 24 m long.

### The cost of painting the four walls of a room 12 m long at ₹ 30 per m2 is ₹ 7560 per m2 and the cost of covering the floor with the mat at ₹ dimensions of the room.

### The dimensions of a room are 14 m x 10 m x 6.5 m There are two doors and 4 windows in the room. Each door measures 2.5 m x 1.2 m and each window measures 1.5 m x 1 m. Find the cost of painting the four walls of the room at ₹ 35 per m2 .

### A 80 m by 64 m rectangular lawn has two roads, each 5 m wide, running through its middle, one parallel to its length and the other parallel to its breadth. Find the cost of gravelling the reads at ₹ 40 per m2 .

### A carpet is laid on floor of a room 8 m by 5 m. There is border of constant width all around the carpet. If the area of the border is 12 m2

### A room 4.9 m long and 3.5 m board is covered with carpet, leaving an uncovered margin of 25 cm all around the room. If the breadth of the carpet is 80 cm, find its cost at ₹ 80 per metre.

### The length and breadth of a rectangular garden are in the ratio 9:5. A path 3.5 m wide, running all around inside it has an area of 1911m2 . Find the dimensions of the garden.

### A footpath of uniform width runs all around the inside of a rectangular field 54m long and 35 m wide. If the area of the path is 420 m2 , find the width of the path.

### A rectangular plot measure 125 m by 78 m. It has gravel path 3 m wide all around on the outside. Find the area of the path and the cost of gravelling it at ₹ 75 per m2

### A rectangular park 358 m long and 18 m wide is to be covered with grass, leaving 2.5 m uncovered all around it. Find the area to be laid with grass.

### The area of rectangle is 192cm2 and its perimeter is 56 cm. Find the dimensions of the rectangle.

### A 36-m-long, 15-m-borad verandah is to be paved with stones, each measuring 6dm by 5 dm. How many stones will be required?

### The floor of a rectangular hall is 24 m long and 18 m wide. How many carpets, each of length 2.5 m and breadth 80 cm, will be required to cover the floor of the hall?

### A room is 16 m long and 13.5 m broad. Find the cost of covering its floor with 75-m-wide carpet at ₹ 60 per metre.

### A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3. The area of the lawn is 3375m2 . Find the cost of fencing the lawn at ₹ 65 per metre.

### The area of a rectangular plot is 462m2is length is 28 m. Find its perimeter

### One side of a rectangle is 12 cm long and its diagonal measure 37 cm. Find the other side and the area of the rectangle.

### The length of a rectangular park is twice its breadth and its perimeter is 840 m. Find the area of the park.

### The perimeter of a rectangular plot of land is 80 m and its breadth is 16 m. Find the length and area of the plot.

### In the given figure, ABC is an equilateral triangle the length of whose side is equal to 10 cm, and ADC is right-angled at D and BD= 8cm. Find the area of the shaded region

### Find the area and perimeter of an isosceles right angled triangle, each of whose equal sides measure 10cm.

### Each of the equal sides of an isosceles triangle measure 2 cm more than its height, and the base of the triangle measure 12 cm. Find the area of the triangle.

### The base of an isosceles triangle measures 80 cm and its area is 360 cm2. Find the perimeter of the triangle.

### Find the length of the hypotenuse of an isosceles right-angled triangle whose area is 200cm2 . Also, find its perimeter

### Find the area of a right – angled triangle, the radius of whose, circumference measures 8 cm and the altitude drawn to the hypotenuse measures 6 cm.

### The base of a right – angled triangle measures 48 cm and its hypotenuse measures 50 cm. Find the area of the triangle.

### If the area of an equilateral triangle is 81

### 11. If the area of an equilateral triangle is 36 Sol: cm2 , find its perimeter.

### The height of an equilateral triangle is 6 cm. Find its area.

### Each side of an equilateral triangle is 10 cm. Find (i) the area of the triangle and (ii) the height of the triangle.

### The length of the two sides of a right triangle containing the right angle differ by 2 cm. If the area of the triangle is 24 c m2 , find the perimeter of the triangle.

### The difference between the sides at the right angles in a right-angled triangle is 7 cm. the area of the triangle is 60 c m2 . Find its perimeter.

### The perimeter of a right triangle is 40 cm and its hypotenuse measure 17 cm. Find the area of the triangle.

### The perimeter of a triangular field is 240m, and its sides are in the ratio 25:17:12. Find the area of the field. Also, find the cost of ploughing the field at ₹ 40 per m2

### The sides of a triangle are in the ratio 5:12:13 and its perimeter is 150 m. Find the area of the triangle.

### Find the area of the triangle whose sides are 18 cm, 24 cm and 30 cm. Also find the height corresponding to the smallest side.

### Find the areas of the triangle whose sides are 42 cm, 34 cm and 20 cm. Also, find the height corresponding to the longest side.

### Find the area of triangle whose base measures 24 cm and the corresponding height measure 14.5 cm.

### In a bulb factory, machines A, B and C manufactures and bulbs respectively. Out of these bulbs and of the bulbs produced respectively by and are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by machine .

Let $A$ : Manufactured from machine $A$, B : Manufactured from machine B C: Manufactured from machine C D : Defective bulb We want to find $P(A \mid D)$, i.e. probability of selected defective bulb...

### An insurance company insured 2000 scooters and 3000 motorcycles. The probability of an accident involving a scooter is , and that of motorcycles is . An insured vehicle met with an accident. Find the probability that the accidented vehicle was a motorcycle.

Let $M$ : Motorcycle S: Scooter A : Accident vechicle We want to find $P(M \mid A)$, i.e. probability of accident vehicle was a motorcycle $\begin{array}{l} P(M \mid A)=\frac{P(M) \cdot P(A \mid...

### A car manufacturing factory has two plants and . Plant manufactures of the cars, and plant , manufactures . At pant of the cars are rated of standard quality, and at plant are rated of standard quality. A car is picked up at random and is found to be of standard quality. A car is picked up at random and is found to be of standard quality. Find the probability that it has come from plant .

Let $X$ : Car produced from plant $X$ $Y$ : Car produced from plant $Y$ S: Car rated as standard quality We want to find $P(X \mid S)$, i.e. selected standard quality car is from plant $X$...

### There are four boxes, and , containing marbles. A contains 1 red, 6 white and 3 black marbles; contains 6 red, 2 white and 2 black marbles; C contains 8 red, 1 white and 1 black marbles; and D contains 6 white and 4 black marbles. One of the boxes is selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from the box ?

Let $A:$ Ball drawn from bag $A$ B: Ball is drawn from bag B $C:$ Ball is drawn from bag $C$ $D:$ Ball is drawn from bag $D$ BB: Black ball WB : White ball RB : Red ball Assuming all boxes have an...

### There are 3 bags, each containing 5 white and 3 black balls. Also, there are 2 bags, each containing 2 white and 4 black balls. A white ball is drawn at random. Find the probability that this ball is from a bag of the first group.

Let $A$ : the set of first 3 bags $B$ : a set of next 2 bags WB : White ball BB : Black ball Now we can change the problem to two bags, i.e. bag A containing 15 white and 9 black balls( 5 white and...

### Urn A contains 7 white and 3 black balls; urn B contains 4 white and 6 black balls; urn C contains 2 white and 8 black balls. One of these urns is chosen at random with probabilities and respectively. From the chosen urn, two balls are drawn at random without replacement. Both the balls happen to be white. Find the probability that the balls are drawn are from urn .

Let $A:$ Ball is drawn from bag $A$ B : Ball is drawn from bag B $C:$ Ball is drawn from bag $C$ BB: Black ball WB: White ball RB: Red ball Probability of picking 2 white balls fro urn $A=\frac{7...

### There are three boxes, the first one containing 1 white, 2 red and 3 black balls; the second one containing 2 white, 3 red and 1 black ball and the third one containing 3 white, 1 red and 2 black balls. A box is chosen at random, and from it, two balls are drawn at random. One ball is red and the other, white. What is the probability that they come from the second box?

let $A:$ Ball drawn from bag $A$ B : Ball is drawn from bag B $C:$ Ball is drawn from bag $C$ BB: Black ball WB: White ball RB: Red ball Assuming, selecting bags is of equal probability i.e....

### Mark the tick against the correct answer in the following:

A.

B.

C.

D.

Solution: Option(C) is correct. To Find: The value of $\tan ^{-1} 2+\tan ^{-1} 3$ Since we know that $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$ $\begin{array}{l}...

### Three urns contain 2 white and 3 black balls; 3 white and 2 black balls, and 4 white and 1 black ball respectively. One ball is drawn from an urn chosen at random, and it was found to be white. Find the probability that it was drawn from the first urn.

let $\mathrm{A}:$ Ball drawn from bag $\mathrm{A}$ $B:$ Ball is drawn from bag $B$ $C:$ Ball is drawn from bag $C$ BB : Black ball WB : White ball Assuming, selecting bags is of equal probability...

### Three urns A, B and C contains 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is chosen at random, and a ball is drawn. If the ball drawn is found to be red, find the probability that the balls was drawn from the first urn .

let $A:$ Ball drawn from bag $A$ B : Ball is drawn from bag B $C:$ Ball is drawn from bag $C$ R: Red ball W : White ball Assuming, selecting bags is of equal probability i.e. $\frac{1}{3}$ We want...

### There are two I and II. The bag I contains 3 white and 4 black balls, and bag II contains 5 white and 6 black balls. One ball is drawn at random from one of the bags and is found to be white. Find the probability that it was drawn from the bag I.

Let $\mathrm{W}$ : White ball B : Black ball $\begin{array}{l} X: 1^{\text {st }} \text { bag } \\ Y: 2^{\text {nd }} \text { bag } \end{array}$ Assuming, selecting bags is of equal probability i.e....

### A bag A contains 1 white and 6 red balls. Another bag contains 4 white and 3 red balls. One of the bags is selected at random, and a ball is drawn from it, which is found to be white. Find the probability that the ball is drawn is from bag . red balls. Another bag contains 4 white and 3 red balls. One of the bags is selected at random, and a ball is drawn from it, which is found to be white. Find the probability that the ball is drawn is from bag A.

Let R : Red ball W : White ball A: Bag A B: Bag B Assuming, selecting bags is of equal probability i.e. $\frac{1}{2}$ We want to find $P(A \mid W)$, i.e. the selected white ball is from bag $A$:...

### Two groups are competing for the positions on the board of directors of a corporation. The probabilities that the first and the second groups will win are and , respectively. Further, if the first group wins, the probability of introducing a new product is , and when the second groups win, the corresponding probability is . Find the probability that the new product introduced was by the second group.

Let $F$ : First group S : Second group $N$ : Introducing a new product We want to find $P(S \mid N)$, i.e. new product introduced by the second group $\begin{array}{l} \mathrm{P}(\mathrm{S} \mid...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D.

Solution: Option(C) is correct. To Find: The value of $\tan ^{-1} 1+\tan ^{-1} \frac{1}{3}$ Let, $x=\tan ^{-1} 1+\tan ^{-1} \frac{1}{3}$ Since we know that $\tan ^{-1} x+\tan ^{-1} y=\tan...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D.

Solution: Option(A) is correct. To Find: The value of $\tan ^{-1}(-1)+\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$ Let, $x=\tan ^{-1}(-1)+\cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)$ $\Rightarrow...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D.

Solution: Option(B) is correct. To Find: The value of $\tan ^{-1} 1+\cos ^{-1}\left(\frac{-1}{2}\right)+\sin ^{-1}\left(\frac{-1}{2}\right)$ Now, let $x=\tan ^{-1} 1+\cos...

### Mark the tick against the correct answer in the following:

A.

B.

C.

D. none of these

Solution: Option(B) is correct. To Find: The value of $\tan ^{-1}(\sqrt{3})-\sec ^{-1}(-2)$ Let, $x=\tan ^{-1}(\sqrt{3})-\sec ^{-1}(-2)$ $\begin{array}{l} \Rightarrow...

### Mark the tick against the correct answer in the following: The value of is

A.

B.

C.

D. none of these

Solution: Option(B) is correct. To Find: The value of $\sin \left(\cos ^{-1} \frac{3}{5}\right)$ Now, let $x=\cos ^{-1} \frac{3}{5}$ $\Rightarrow \cos x=\frac{3}{5}$ Now , $\sin x=\sqrt{1-\cos ^{2}...

### Mark the tick against the correct answer in the following: The value of is

A.

B.

C.

D. none of these

Solution: Option(A) is correct. To Find: The value of $\sec ^{-1}\left(\sec \left(\frac{8 \pi}{5}\right)\right)$ Now, let $x=\sec ^{-1}\left(\sec \left(\frac{8 \pi}{5}\right)\right)$ $\Rightarrow...

### Mark the tick against the correct answer in the following: The value of is

A.

B.

C.

D. none of these

Solution: To Find: The value of $\tan ^{-1}\left(\tan \left(\frac{7 \pi}{6}\right)\right)$ Now, let $x=\tan ^{-1}\left(\tan \left(\frac{7 \pi}{6}\right)\right)$ $\Rightarrow \tan x=\tan...

### Mark the tick against the correct answer in the following: The value of is

A.

B.

C.

D. none of these

Solution: Option(C) us correct. To Find: The value of $\sin ^{-1}\left(\sin \left(\frac{2 \pi}{3}\right)\right)$ Now, let $x=\sin ^{-1}\left(\sin \left(\frac{2 \pi}{3}\right)\right)$ $\Rightarrow...

### Mark the tick against the correct answer in the following: The principal value of is

A.

B.

C.

D. none of these

Solution: Option(A) is correct. To Find: The Principle value of $\operatorname{cosec}^{-1}(-\sqrt{2})$ Let the principle value be given by $\mathrm{x}$ Now, let...

### Draw two concentric circles of radii 4 cm and 6 cm. Construct a tangent to the smaller circle from a point on the larger circle. Measure the length of this tangent.

### Draw a circle of radius 4 cm. Draw tangent to the circle making an angle of 60 with a line passing through the centre.

### Draw a circle of radius 3.5 cm. Draw a pair of tangents to this circle which are inclined to each other at an angle of 60 . Write the steps of construction.

### Draw a circle of radius 4.8 cm. Take a point P on it. Without using the centre of the circle, construct a tangent at the point P. Write the steps of construction.

### Draw a ABC , right-angled at B such that AB = 3 cm and BC = 4cm. Now, Construct a triangle

### Construct an isosceles triangle whose base is 9 cm and altitude 5cm. Construct another

### Construct a ABC in which BC = 5cm, C 60 and altitude from A equal to 3 cm. Construct

### Construct a ABC Sol: in which B= 6.5 cm, AB = 4.5 cm and ABC 60

### Draw a line segment AB of length 6.5 cm and divided it in the ratio 4 : 7. Measure each of the two parts.

### Draw a line segment AB of length 5.4 cm. Divide it into six equal parts. Write the steps of construction.

### Construct a tangent to a circle of radius 4 cm form a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation.

### Draw a circle of radius 32 cm. Draw a tangent to the circle making an angle 30 with a line passing through the centre.

### Write the steps of construction for drawing a pair of tangents to a circle of radius 3 cm , which are inclined to each other at an angle of 60 .

### Draw a circle of radius 4.2. Draw a pair of tangents to this circle inclined to each other at an angle of 45

### Draw a line segment AB of length 8 cm. Taking A as centre , draw a circle of radius 4 cm and taking B as centre , draw another circle of radius 3 cm. Construct tangents to each circle form the centre of the other circle.

### Draw a circle with the help of a bangle. Take any point P outside the circle. Construct the pair of tangents form the point P to the circle

### Draw a circle with center O and radius 4 cm. Draw any diameter AB of this circle. Construct tangents to the circle at each of the two end points of the diameter AB.

### Draw a circle of radius 3.5 cm. Take two points A and B on one of its extended diameter, each at a distance of 5 cm from its center. Draw tangents to the circle from each of these points A and B.

### 2. Draw two tangents to a circle of radius 3.5 cm form a point P at a distance of 6.2 cm form its centre.

### Draw a circle of radius 3 cm. Form a point P, 7 cm away from the centre of the circle, draw two tangents to the circle. Also, measure the lengths of the tangents.

### Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3

### Construct an isosceles triangles whose base is 8 cm and altitude 4 cm and then another

### To construct a triangle similar to

### Find the eccentricity of an ellipse whose latus rectum is one half of its major axis.

### Find the eccentricity of an ellipse whose latus rectum is one half of its minor axis.

### Find the equation of an ellipse whose eccentricity is

, the latus rectum is

, and the center is at the origin.

### Find the equation of an ellipse, the lengths of whose major and mirror axes are

and

units respectively.

### Find the equation of the ellipse which passes through the point

and having its foci at

.

### Find the equation of the ellipse with eccentricity

, foci on the y-axis, center at the origin and passing through the point

.

Given Eccentricity = \[\frac{3}{4}\] We know that Eccentricity = c/a Therefore,c=\[\frac{3}{4}\]a

### Find the equation of the ellipse with center at the origin, the major axis on the x-axis and passing through the points

and

.

Given: Center is at the origin and Major axis is along x – axis So, Equation of ellipse is of the form \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]…(i) Given that ellipse passing...

### Find the equation of the ellipse whose foci are at

and

Given: Coordinates of foci = \[\left( \mathbf{0},\text{ }\pm \mathbf{4} \right)\] …(i) We know that, Coordinates of foci = \[\left( 0,\text{ }\pm c \right)\] …(ii) The coordinates of the foci are...

### Find the equation of the ellipse whose foci are at

and e=1/2

Let the equation of the required ellipse be \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] Given: Coordinates of foci = \[\left( \pm 1,\text{ }0 \right)\] …(i) We know that,...

### Find the equation of the ellipse whose foci are

and the eccentricity is

Let the equation of the required ellipse be Given: Coordinates of foci = \[\left( \pm 2,\text{ }0 \right)\]…(iii) We know that, Coordinates of foci = \[\left( \pm c,\text{ }0 \right)\]…(iv) ∴ From...

### Find the equation of the ellipse the ends of whose major and minor axes are

and

respectively.

Given: Ends of Major Axis = \[\left( \pm \mathbf{4},\text{ }\mathbf{0} \right)\] and Ends of Minor Axis = \[\left( \mathbf{0},\text{ }\pm \mathbf{3} \right)\] Here, we can see that the major axis is...

### Find the equation of the ellipse whose vertices are the

and foci at

.

Given: Vertices = \[(0,\pm 4)\] …(i) The vertices are of the form = (0, ±a) …(ii) Hence, the major axis is along y – axis ∴ From eq. (i) and (ii), we get \[\begin{array}{*{35}{l}} a\text{ }=\text{...

### Find the equation of the ellipse whose vertices are at

and foci at

.

Given: Vertices = \[\left( \pm \mathbf{6},\text{ }\mathbf{0} \right)\] …(i) The vertices are of the form = \[\left( \pm a,\text{ }0 \right)\] …(ii) Hence, the major axis is along x – axis ∴ From eq....

### Find the (v) length of the latus rectum of each of the following ellipses.

Given: \[\mathbf{25}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\] Divide by \[100\] to both the sides, we get...

### Find the (iii) coordinates of the foci, (iv) eccentricity

Given: \[\mathbf{25}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\] Divide by \[100\] to both the sides, we get...

### Find the (i) lengths of major axes, (ii) coordinates of the vertices

Given: \[\mathbf{25}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{4}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{100}\] Divide by \[100\] to both the sides, we get...

### Find the (v) length of the latus rectum of each of the following ellipses.

Given: \[\mathbf{16}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{16}\] Divide by \[16\] to both the sides, we get...

### Find the (iii) coordinates of the foci, (iv) eccentricity

Given: \[\mathbf{16}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{16}\] Divide by \[16\] to both the sides, we get...

### Find the (i) lengths of major axes, (ii) coordinates of the vertices

Given: \[\mathbf{16}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{16}\] Divide by \[16\] to both the sides, we get...

### Prove that:

Answer: = cot x = RHS ∴ LHS = RHS Hence Proved

### Find the (v) length of the latus rectum of each of the following ellipses.

Given: \[\mathbf{9}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{36}\] Divide by \[36\] to both the sides, we get \[\frac{9}{36}{{x}^{2}}+\frac{1}{36}{{y}^{2}}=1\]...

### Find the (iii) coordinates of the foci, (iv) eccentricity

Given: \[\mathbf{9}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{36}\] Divide by \[36\] to both the sides, we get \[\frac{9}{36}{{x}^{2}}+\frac{1}{36}{{y}^{2}}=1\]...

### Find the (i) lengths of major axes, (ii) coordinates of the vertices

Given: \[\mathbf{9}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{36}\] Divide by \[36\] to both the sides, we get \[\frac{9}{36}{{x}^{2}}+\frac{1}{36}{{y}^{2}}=1\]...

### Prove that:

Answer: = tan x = RHS ∴ LHS = RHS Hence Proved

### Find the (v) length of the latus rectum of each of the following ellipses.

Given: \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{18}\]…(i) Divide by \[18\] to both the sides, we get...

### Prove that:

Answer: Taking LHS = cos x + sin x = RHS ∴ LHS = RHS Hence Proved

### Find the (iii) coordinates of the foci, (iv) eccentricity

Given: \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{18}\]…(i) Divide by \[18\] to both the sides, we get...

### Find the (i) lengths of major axes, (ii) coordinates of the vertices

Given: \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{18}\]…(i) Divide by \[18\] to both the sides, we get...

### If cos x = -1/3 , find the value of cos 3x

Answer; We know that, cos 3x = 4cos3 x – 3 cosx Putting the values, we get

### If sinx = 1/6, find the value of sin 3x.

Answer: To find: sin 3x We know that, sin 3x = 3 sinx – sin3 x Putting the values, we get

### Find the (v) length of the latus rectum of each of the following ellipses.

Answer :

Given: \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{16}=1\]….(i) Since, \[9<16\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\]…(ii) Comparing eq. (i)...

### Find the (iii) coordinates of the foci, (iv) eccentricity

Given: \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{16}=1\]….(i) Since, \[9<16\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\]…(ii) Comparing...

### Find the (i) lengths of major axes, (ii) coordinates of the vertices

Given: \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{16}=1\]….(i) Since, \[9<16\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\]…(ii) Comparing eq. (i)...

### If , find the values of tan 2x

Answer: We know that:

### Find the (v) length of the latus rectum of each of the following ellipses.

Given: \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{25}=1\]…(i) Since, \[4\text{ }<\text{ }25\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\]…(ii)...

### Find the (iii) coordinates of the foci, (iv) eccentricity

Given: \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{25}=1\]…(i) Since, \[4\text{ }<\text{ }25\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\]…(ii)...

### If , find the values of cos 2x

Answer:

### Find the (i) lengths of major axes, (ii) coordinates of the vertices

Given: \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{25}=1\]…(i) Since, \[4\text{ }<\text{ }25\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\]…(ii)...

### If , find the values of sin 2x

Answer: We know that

### Find the (v) length of the latus rectum of each of the following ellipses.

Given: \[\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{9}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{1}\] \[\frac{{{x}^{2}}}{\frac{1}{4}}+\frac{{{y}^{2}}}{\frac{1}{9}}=1\]….(i) Since,...

### Find the (iii) coordinates of the foci, (iv) eccentricity

Given: \[\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}+\text{ }\mathbf{9}{{\mathbf{y}}^{\mathbf{2}}}=\text{ }\mathbf{1}\] \[\frac{{{x}^{2}}}{\frac{1}{4}}+\frac{{{y}^{2}}}{\frac{1}{9}}=1\]….(i) Since,...